Supercells - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 214

Figure 4.32. Idealized example of a straight (unidirectional) hodograph (thin solid line), which
is labeled every kilometer. The wind vectors at 0 and 2 km AGL are shown for purposes of
illustration.

vertical velocity is nonzero; the speed of ascending air (e.g., along fronts, outflow
boundaries, etc.), however ( 10 cm s 1 -1m s 1 ), is an order of magnitude or more
less than that of buoyant updrafts ( 10m s 1 ).
Thus, using (4.42)-(4.44), it is seen that (4.40) may be expressed as

2 p 0 d ¼½ð@

u 0

2

þð@v 0

2

w 0

2

0 r =@ x Þ =@ y Þ þð@ =@ z Þ

u 0 @v 0 w 0 u 0 w 0 @v 0

2 ½ð@ =@ y =@ x þ@ =@ x @ =@ z þ@ =@ y =@ z Þ

w 0 w 0 2 ½@ =@ x @ U =@ z þ@ =@ y @ V =@ z ð 4 : 45 Þ

The terms on the RHS are separated into the following linear terms

w 0 w 0 w 0 2 ½@ =@ x @ U =@ z þ@ =@ y @ V =@ z ¼ 2 @ V =@ z ð 4 : 46 Þ

EJ

and nonlinear terms

u 0

2

þð@v 0

2

w 0

2

½ð@ =@ x Þ =@ y Þ þð@ =@ z Þ

u 0 @v 0 w 0 u 0 w 0 @v 0

2 ð@ =@ y =@ x þ@ =@ x @ =@ z þ@ =@ y =@ z Þ

as first shown by Rich Rotunno and Joe Klemp in the mid-1980s. The nonlinear
terms ½ð@ u 0

2

þð@v 0

2

w 0

2

=@

x Þ

=@

y Þ

þð@ =@

z Þ

are called ''fluid extension terms''

u 0

@v 0

w 0 u 0 terms ð@

=@ =@

x þ@

=@ @ =@

z þ

(as in (4.30)) and the nonlinear y x

w 0 @v 0 @ =@ y =@ z Þ are called ''shear terms'' (as in (4.30)).